In this paper we prove, by showing that solutions have exactly the same degree of regularity as the obstacle, optimal regularity results for obstacle problems involving evolutionary $p$-Laplace type operators. A main ingredient, of independent interest, is a new intrinsic interpolative geometry allowing for optimal linearization principles via blow-up analysis at contact points. This also opens the way to the proof of a removability theorem for solutions to evolutionary $p$-Laplace type equations. A basic feature of the paper is that no differentiability in time is assumed on the obstacle; this is in line with the corresponding linear results